This paper deals with a class of time-periodic Hamiltonian systems obtained by a time-dependent perturbation from an autonomous system with a singularity at q = 0 in configuration space. It is shown that, T being the period of the perturbation, nondegenerate families of T-periodic orbits in the unperturbed problem branch off into a certain number of T-periodic orbits for the perturbed problem.

We show that the problem

has a unique solution and that the asymptotic behaviour of this solution as x → −∞ is given by w(x) = x + B ∣x∣−1/4 cos (b ∣x∣3/2 + φ) + O(x−1), for some constant φ. Here B ≠ 0 depends on ε and In particular there are no even solutions of problem (*) for any ε > 0.

We consider two boundary value problems (of Neumann or related type) associated with the equation in Ω. The existence of a solution was previously established assuming that p < N/(N −s2). (N dimension of Ω.) We prove that this exponent is critical for these problems, at least in the radially symmetric case when Ω is a ball. This is understood in the sense that the existence result does not hold when p ≧ N/(N − 2).

Consider the second order Hamiltonian system:

where q ∊ ℝn and V ∊ C1 (ℝ ×ℝn ℝ) is T periodic in t. Suppose Vq (t, 0) = 0, 0 is a local maximum for V(t,.) and V(t, x) | x| → ∞ Under these and some additional technical assumptions we prove that (HS) has a homoclinic orbit q emanating from 0. The orbit q is obtained as the limit as k → ∞ of 2kT periodic solutions (i.e. subharmonics) qk of (HS). The subharmonics qk are obtained in turn via the Mountain Pass Theorem.

Let Mn be a manifold supposed “nicely curved” isometrically immersed in ℝn+p. Starting from a generalised Gauss map associated to the splitting of the normal bundle defined from the values of the fundamental forms of M of order k (k ≧ 0), we give necessary and sufficient conditions for the map to be totally geodesic and harmonic . For k = 0 is the classical Gauss map and our formula reduces to Ruh–Vilm's formula with a more precise formulation due to the consideration of the splitting of the normal bundle.

We also give necessary conditions for M, supposed complete, to admit an isometric immersion with . This theorem generalises a theorem of Vilms on the manifolds with second fundamental forms parallel (case k = 0). The result is interesting as the class of manifolds satisfying the condition is larger than the class of manifolds satisfying .

It is shown that a cantilevered beam with weak viscoelastic damping of Boltzmann-type can be uniformly stabilised by velocity feedback applied as a shearing force at the free end of the beam. Estimates for the viscoelastic energy are derived using the energy multiplier method. The energy decay is related to the decay of the relaxation modulus associated with the viscoelastic material.

We examine slow viscous flow past a concentrated bed of small stationary viscous bubbles of a second fluid, and derive Darcy's law relating the average fluid velocity to the overall pressure gradient and body force.

The paper suggests a procedure for direct construction of minimal extensions of constrained optimisation problems, particularly those containing controls in coefficients of elliptic equations. The preliminary version of the procedure has been described in [1].

A characterisation is obtained of all the regularly solvable operators and their adjoints generated by a general differential expression in . The domains of these operators are described in terms of boundary conditions involving the solutions of Mu = λwu and the adjoint equation . The results include those of Sun Jiong [15] concerning self-adjoint realisations of a symmetric M when the minimal operator has equal deficiency indices: if the deficiency indices are unequal the maximal symmetric operators are determined by the results herein. Another special case concerns the J -self-adjoint operators, where J denotes complex conjugation, and for this we recover the results of Zai-jiu Shang in [16].

The authors prove results on uniqueness and global existence of initial and boundary value problems for the nonlinear pseudoparabolic equation

with nonhomogeneous boundary conditions. A salient feature of the paper is that F and its partial derivatives are allowed to be unbounded. In the special case b(x, t)= α2 (a positive constant), it is proved that the corresponding solution uα, under appropriate conditions on the data (which are satisfied, for example, by the Benjamin–Bona–Mahony equation), uα→ u0 the solution corresponding to β = 0, on sufficiently small time interval. A result on the asymptotic behaviour of the solution is given for t → ∞.

We study the rank one convexity of some functions f(ξ) where ξ is a 2 × 2 matrix. Examples such as |ξ|2α + h(detξ) and | ξ |2α (| ξ |2 − γdet ξ) are investigated. Numerical computations are done on the example of Dacorogna and Marcellini, indicating that this function is quasiconvex.

We consider a system of integrodifferential equations

where f(t, x) and F(t, s, x, y) are almost periodic in t uniformly for parameters, and we assume that the system has a bounded solution u(t). To discuss the existence of an almost periodic solution, we consider the relationship between the total stability of u(t) with respect to a certain metric ρ and the separation condition with respect to ρ. Moreover, we discuss a sufficient condition for the existence of a positive almost periodic solution of a model of the dynamics of an n-species system.